A friendly iteration forcing that the four cardinal characteristics of $\mathcal{E}$ can be pairwise different

TL;DR

This paper demonstrates, through a specialized forcing technique, that the four cardinal characteristics of the ideal of measure-zero sets can be made pairwise distinct, advancing understanding of their possible relationships.

Contribution

It introduces a novel forcing method using an ultrafilter-extendable matrix iteration to separate the cardinal characteristics of the ideal of measure-zero sets.

Findings

The additivity of $\

The covering number of $\

The cofinality of $\

Abstract

Let $\mathcal{E}$ be the $\sigma$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different.